# 1 st Grade Mathematics Unpacked Content For the new Common Core State Standards that will be effective in all North Carolina schools in the

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1 1 st Grade Mathematics Unpacked Content For the new Common Core State Standards that will be effective in all North Carolina schools in the This document is designed to help North Carolina educators teach the Common Core State Standards (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. What is the purpose of this document? To increase student achievement by ensuring educators understand specifically what the new standards mean a student must know, understand and be able to do. This document may also be used to facilitate discussion among teachers and curriculum staff and to encourage coherence in the sequence, pacing, and units of study for grade-level curricula. This document, along with on-going professional development, is one of many resources used to understand and teach the CCSS. What is in the document? Descriptions of what each standard means a student will know, understand and be able to do. The unpacking of the standards done in this document is an effort to answer a simple question What does this standard mean that a student must know and be able to do? and to ensure the description is helpful, specific and comprehensive for educators. How do I send Feedback? We intend the explanations and examples in this document to be helpful and specific. That said, we believe that as this document is used, teachers and educators will find ways in which the unpacking can be improved and made ever more useful. Please send feedback to us at and we will use your input to refine our unpacking of the standards. Thank You! Just want the standards alone? You can find the standards alone at Updated: August st Grade Mathematics Unpacked Content

9 Number Balance A student uses a number balance to investigate the commutative property. If 8 and 2 equals 10, then I think that if I put a weight on 2 first this time and then on 8, it ll also be 10. Associative Property Examples: Number Line: = Student A: First I jumped to 5. Then, I jumped 4 more, so I landed on 9. Then I jumped 5 more and landed on 14. Student B: I got 14, too, but I did it a different way. First I jumped to 5. Then, I jumped 5 again. That s 10. Then, I jumped 4 more. See, 14! Mental Math: There are 9 red jelly beans, 7 green jelly beans, and 3 black jelly beans. How many jelly beans are there in all? Student: I know that is 10. And 10 and 9 is 19. There are 19 jelly beans. 1st Grade Mathematics Unpacked Content page10

11 Student B: I put 9 counters on the 10 frame. Just looking at it I can tell that I need 1 more to get to 10. Then I need 5 more to get to 15. So, I need 6 counters. *Back Down Through 10 This strategy uses take-away and 10 as a bridge. Students take away an amount to make 10, and then take away the rest. It is helpful for facts where the ones digit of the two-digit number is close to the number being subtracted. Example: 16 7 = Student A: I ll start with 16 and take off 6. That makes 10. I ll take one more off and that makes = 9. Student B: I used 16 counters to fill one ten frame completely and most of the other one. Then, I can take these 6 off from the 2 nd ten frame. Then, I ll take one more from the first ten frame. That leaves 9 on the ten frame. *Van de Walle & Lovin, st Grade Mathematics Unpacked Content page12

16 Common Core Cluster Understand place value. Students develop, discuss, and use efficient, accurate, and generalizable methods to add within 100 and subtract multiples of 10. They compare whole numbers (at least to 100) to develop understanding of and solve problems involving their relative sizes. They think of whole numbers between 10 and 100 in terms of tens and ones (especially recognizing the numbers 11 to 19 as composed of a ten and some ones). Through activities that build number sense, they understand the order of the counting numbers and their relative magnitudes. Mathematically proficient students communicate precisely by engaging in discussion about their reasoning using appropriate mathematical language. The terms students should learn to use with increasing precision with this cluster are: tens, ones, bundle, left-overs, singles, groups, greater/less than, equal to Common Core Standard Unpacking 1.NBT.2 Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases: a. 10 can be thought of as a bundle of ten ones called a ten. What do these standards mean a child will know and be able to do? First Grade students are introduced to the idea that a bundle of ten ones is called a ten. This is known as unitizing. When First Grade students unitize a group of ten ones as a whole unit ( a ten ), they are able to count groups as though they were individual objects. For example, 4 trains of ten cubes each have a value of 10 and would be counted as 40 rather than as 4. This is a monumental shift in thinking, and can often be challenging for young children to consider a group of something as one when all previous experiences have been counting single objects. This is the foundation of the place value system and requires time and rich experiences with concrete manipulatives to develop. A student s ability to conserve number is an important aspect of this standard. It is not obvious to young children that 42 cubes is the same amount as 4 tens and 2 left-overs. It is also not obvious that 42 could also be composed of 2 groups of 10 and 22 leftovers. Therefore, first graders require ample time grouping proportional objects (e.g., cubes, beans, beads, ten-frames) to make groups of ten, rather than using pre-grouped materials (e.g., base ten blocks, pre-made bean sticks) that have to be traded or are non-proportional (e.g., money). Example: 42 cubes can be grouped many different ways and still remain a total of 42 cubes. We want children to construct the idea that all of these are the same and that the sameness is clearly evident by virtue of the groupings of ten. Groupings by tens is not just a rule that is followed but that any grouping by tens, including all or some of the singles, can help tell how many. (Van de Walle & Lovin, p. 124) 1st Grade Mathematics Unpacked Content page17

17 As children build this understanding of grouping, they move through several stages: Counting By Ones; Counting by Groups & Singles; and Counting by Tens and Ones. Counting By Ones: At first, even though First Graders will have grouped objects into tens and left-overs, they rely on counting all of the individual cubes by ones to determine the final amount. It is seen as the only way to determine how many. Example: Teacher:Howmanycountersdoyouhave? Student:1,2,3,4,.41,42.Ihave42counters. Counting By Groups and Singles: While students are able to group objects into collections of ten and now tell how many groups of tens and left-overs there are, they still rely on counting by ones to determine the final amount. They are unable to use the groups and left-overs to determine how many. Example: Teacher:Howmanycountersdoyouhave? Student:Ihave4groupsoftenand2left overs. Teacher:Doesthathelpyouknowhowmany?Howmanydoyouhave? Student:Letmesee.1,2,3,4,5,.41,42.Ihave42counters. Counting by Tens & Ones: Students are able to group objects into ten and ones, tell how many groups and leftovers there are, and now use that information to tell how many. Ex: I have 3 groups of ten and 4 left-overs. That means that there are 34 cubes in all. Occasionally, as this stage is becoming fully developed, first graders rely on counting by ones to really know that there are 34, even though they may have just counted the total by groups and left-overs. Example: Teacher:Howmanycountersdoyouhave? Student:Ihave4groupsoftenand2left overs. Teacher:Doesthathelpyouknowhowmany?Howmanydoyouhave? Student:Yes.ThatmeansthatIhave42counters. Teacher:Areyousure? Student:Um.Letmecountjusttomakesure 1,2,3, 41,42.Yes.Iwasright. Thereare42counters. 1st Grade Mathematics Unpacked Content page18

19 c. The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones). 1.NBT.3 Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <. First Grade students apply their understanding of groups of ten as stated in 1.NBT.2b to decade numbers (e.g. 10, 20, 30, 40). As they work with groupable objects, first grade students understand that 10, 20, 30 80, 90 are comprised of a certain amount of groups of tens with none left-over. First Grade students use their understanding of groups and order of digits to compare two numbers by examining the amount of tens and ones in each number. After numerous experiences verbally comparing two sets of objects using comparison vocabulary (e.g., 42 is more than is less than 52, 61 is the same amount as 61.), first grade students connect the vocabulary to the symbols: greater than (>), less than (<), equal to (=). Example: Compare these two numbers StudentA 42has4tensand2ones.45has4tensand5 ones.theyhavethesamenumberoftens,but45 hasmoreonesthan42.so,42islessthan45. 42<45 StudentB 42islessthan45.IknowthisbecausewhenI countupisay42beforeisay45. 42<45 Thissays42islessthan45. 1st Grade Mathematics Unpacked Content page20

21 Example: 63 apples are in the basket. Mary put 20 more apples in the basket. How many apples are in the basket? Student A: I used ten frames. I picked out 6 filled ten frames. That s 60. I got the ten frame with 3 on it. That s 63. Then, I picked one more filled ten frame for part of the 20 that Mary put in. That made 73. Then, I got one more filled ten frame to make the rest of the 20 apples from Mary. That s 83. So, there are 83 apples in the basket = =83 Student B: I used a hundreds chart. I started at 63 and jumped down one row to 73. That means I moved 10 spaces. Then, I jumped down one more row (that s another 10 spaces) and landed on 83. So, there are 83 apples in the basket = =83 Student C: I knew that 10 more than 63 is 73. And 10 more than 73 is 83. So, there are 83 apples in the basket = =83 1st Grade Mathematics Unpacked Content page22

22 1.NBT.5 Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used. First Graders build on their county by tens work in Kindergarten by mentally adding ten more and ten less than any number less than 100. First graders are not expected to compute differences of two-digit numbers other than multiples of ten. Ample experiences with ten frames and the number line provide students with opportunities to think about groups of ten, moving them beyond simply rote counting by tens on and off the decade. Such representations lead to solving such problems mentally. Example: There are 74 birds in the park. 10 birds fly away. How many birds are in the park now? Student A I thought about a number line. I started at 74. Then, because 10 birds flew away, I took a leap of 10. I landed on 64. So, there are 64 birds left in the park. Student B I pictured 7 ten frames and 4 left over in my head. Since 10 birds flew away, I took one of the ten frames away. That left 6 ten frames and 4 left over. So, there are 64 birds left in the park. Student C I know that 10 less than 74 is 64. So there are 64 birds in the park. 1st Grade Mathematics Unpacked Content page23

23 1.NBT.6 Subtract multiples of 10 in the range from multiples of 10 in the range (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. First Grade students use concrete models, drawings and place value strategies to subtract multiples of 10 from decade numbers (e.g., 30, 40, 50). They often use similar strategies as discussed in 1.OA.4. Example: There are 60 students in the gym. 30 students leave. How many students are still in the gym? Student A I used a number line. I started at 60 and moved back 3 jumps of 10 and landed on 30. There are 30 students left = = =30 Student B I used ten frames. I had 6 ten frames- that s 60. I removed three ten frames because 30 students left the gym. There are 30 students left in the gym =30 Student C I thought, 30 and what makes 60?. I know 3 and 3 is 6. So, I thought that 30 and 30 makes 60. There are 30 students still in the gym =60 1st Grade Mathematics Unpacked Content page24

24 Measurement and Data 1.MD Common Core Cluster Measure lengths indirectly and by iterating length units. Students develop an understanding of the meaning and processes of measurement, including underlying concepts such as iterating (the mental activity of building up the length of an object with equal-sized units) and the transitivity principle for indirect measurement. 1 1 Students should apply the principle of transitivity of measurement to make indirect comparisons, but they need not use this technical term. Mathematically proficient students communicate precisely by engaging in discussion about their reasoning using appropriate mathematical language. The terms students should learn to use with increasing precision with this cluster are: measure, order, length, height, more, less, longer than, shorter than, first, second, third, gap, overlap, about, a little less than, a little more than Common Core Standard Unpacking What do these standards mean a child will know and be able to do? 1.MD.1 Order three objects First Grade students continue to use direct comparison to compare lengths. Direct comparison means that students compare by length; compare the the amount of an attribute in two objects without measurement. lengths of two objects indirectly by using a third Example: Who is taller? object. Student: Let s stand back to back and compare our heights. Look! I m taller! Example: Find at least 3 objects in the classroom that are the same length as, longer than, and shorter than your forearm. Sometimes, a third object can be used as an intermediary, allowing indirect comparison. For example, if we know that Aleisha is taller than Barbara and that Barbara is taller than Callie, then we know (due to the transitivity of taller than ) that Aleisha is taller than Callie, even if Aleisha and Callie never stand back to back. This concept is referred to as the transitivity principle for indirect measurement. Example: The snake handler is trying to put the snakes in order- from shortest to longest. She knows that the red snake is longer than the green snake. She also knows that the green snake is longer than the blue snake. What order should she put the snakes? Student: Ok. I know that the red snake is longer than the green snake and the blue snake because, since it s longer than the green, that means that it s also longer than the blue snake. So the longest snake is the red snake. I also know that the green snake and red snake are both longer than the blue snake. So, the blue snake is the shortest snake. That means that the green snake is the medium sized snake. 1st Grade Mathematics Unpacked Content page25

25 NOTE: The Transitivity Principle ( transitivity ) 1 : If the length of object A is greater than the length of object B, and the length of object B is greater than the length of object C, then the length of object A is greater than the length of object C. This principle applies to measurement of other quantities as well. Example: Which is longer: the height of the bookshelf or the height of a desk? Student A: I used a pencil to measure the height of the bookshelf and it was 6 pencils long. I used the same pencil to measure the height of the desk and the desk was 4 pencils long. Therefore, the bookshelf is taller than the desk. Student B: I used a book to measure the bookshelf and it was 3 books long. I used the same book to measure the height of the desk and it was a little less than 2 books long. Therefore, the bookshelf is taller than the desk. Another important set of skills and understandings is ordering a set of objects by length. Such sequencing requires multiple comparisons (no more than 6 objects). Students need to understand that each object in a seriation is larger than those that come before it, and shorter than those that come after. Example: The snake handler is trying to put the snakes in order- from shortest to longest. Here are the three snakes (3 strings of different length and color). What order should she put the snakes? Student: Ok. I will lay the snakes next to each other. I need to make sure to be careful and line them up so they all start at the same place. So, the blue snake is the shortest. The green snake is the longest. And the red snake is medium-sized. So, I ll put them in order from shortest to longest: blue, red, green. (Progressions for CCSSM: Geometric Measurement, The CCSS Writing Team, June 2012.) 1.MD.2 Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps. Limit to contexts where the object being measured is spanned by a whole number of length units with no gaps or overlaps. First Graders use objects to measure items to help students focus on the attribute being measured. Objects also lends itself to future discussions regarding the need for a standard unit. First Grade students use multiple copies of one object to measure the length larger object. They learn to lay physical units such as centimeter or inch manipulatives end-to-end and count them to measure a length. Through numerous experiences and careful questioning by the teacher, students will recognize the importance of careful measuring so that there are not any gaps or overlaps in order to get an accurate measurement. This concept is a foundational building block for the concept of area in 3 rd Grade. Example: How long is the pencil, using paper clips to measure? Student: I carefully placed paper clips end to end. The pencil is 5 paper clips long. I thought it would take about 6 paperclips. 1st Grade Mathematics Unpacked Content page26

26 When students use different sized units to measure the same object, they learn that the sizes of the units must be considered, rather than relying solely on the amount of objects counted. Example: Which row is longer? Student Incorrect Response: The row with 6 sticks is longer. Row B is longer. Student Correct Response: They are both the same length. See, they match up end to end. In addition, understanding that the results of measurement and direct comparison have the same results encourages children to use measurement strategies. Example: Which string is longer? Justify your reasoning. Student: I placed the two strings side by side. The red string is longer than the blue string. But, to make sure, I used color tiles to measure both strings. The red string measured 8 color tiles. The blue string measure 6 color tiles. So, I was right. The red string is longer. NOTE: The instructional progression for teaching measurement begins by ensuring that students can perform direct comparisons. Then, children should engage in experiences that allow them to connect number to length, using manipulative units that have a standard unit of length, such as centimeter cubes. These can be labeled length-units with the students. Students learn to lay such physical units end-to-end and count them to measure a length. They compare the results of measuring to direct and indirect comparisons. (Progressions for CCSSM: Geometric Measurement, The CCSS Writing Team, June 2012.) 1st Grade Mathematics Unpacked Content page27

30 the shares. Understand for these examples that decomposing into more equal shares creates smaller shares. Example: How can you and a friend share equally (partition) this piece of paper so that you both have the same amount of paper to paint a picture? Student1 Iwouldsplitthepaperrightdownthemiddle.Thatgivesus 2halves.Ihavehalfofthepaperandmyfriendhasthe otherhalfofthepaper. Student2 Iwouldsplititfromcornertocorner(diagonally). ShegetshalfofthepaperandIgethalfofthepaper. See,ifwecutontheline,thepartsarethesamesize. Example: Let s take a look at this pizza. Teacher: There is pizza for dinner. What do you notice about the slices on the pizza? Student: There are two slices on the pizza. Each slice is the same size. Those are big slices! Teacher: If we cut the same pizza into four slices (fourths), do you think the slices would be the same size, larger, or smaller as the slices on this pizza? Student: When you cut the pizza into fourths, the slices are smaller than the other pizza. More slices mean that the slices get smaller and smaller. I want a slice from that first pizza! 1st Grade Mathematics Unpacked Content page32

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